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Rocket Calculator
Back to Math And Terraforming. This page is meant to calculate the size and power needed for a spaceship. It is made for rockets that exist today, but can easily be adapted for future ships. The formulas listed here are made in a way that is compatible with a Microsoft Excel format. Basic Data The calculations need to be done for each stage of a ship, repeatedly. Some factors can be determined from others. The following parameters need to be calculated or listed for each stage. Stage Parameters A rocket is made of many stages. We need to know for each stage its parameters. #'Total mass (Mt)' is the mass of the stage, with all fuel, plus the mass of the upper stages. ##Mt = Mts + Mp. #'Dry mass (Md)' is the mass of the stage but without fuel (the fuel canisters) plus the upper stages. ##Md = Mt - Mf. ##Md = Mts + Mp. #'Fuel mass (Mf)' is the mass of fuel available for this stage. ##Mf = Mt - Md. ##Mf = Mts - Mds. ##Mf = Mts * Teff. #'Stage mass (Mts)' is the mass of the current stage. ##Mts = Mds + Mf. #'Dry stage mass (Mds)' is the mass of the current stage. ##Mds = Mts - Mf. #'Payload mass (Mp)' is the mass of the upper stages. For chemical rockets, it usually is 0.9. #'Stage fuel efficiency (Teff)' is the fuel storage efficiency of a stage. ##Teff = Mts/(Mts-Mds). Engine Parameters These parameters define the engine. #'Engine specific impulse (Isp)' is a constant that defines the efficiency of an engine. For old engines, Isp is around 300. For modern chemical engines, Isp is slightly above 400. For ion engines, Isp is higher then 2000. A higher Isp means that the engine will produce more thrust with less fuel. #'Number of engines' is an important factor. #'Engine thrust' is the power of an engine. It is described in Newtons (N). Chemical engines usually have thousands of kN, while ion engines generate less then 1 N. #'Burn speed' defines how fast the engine is using its propellant. ##Burn speed = (engine thrust*number of engines)/(Isp*9.81). #'Time to burn all fuel' shows how much time an engine will need to consume all propellant. ##Time to burn all fuel = fuel volume / burn speed. #'Gravity (G)' is the force you need to fight with during a liftoff. In these calculations, for Earth, G = 1. #'Thrust to weight (TWR)' shows if engine's thrust is enough to counter gravity. If the value is below 1, the ship is going down. ##TWR = (nr of engines*engine thrust)/(ship mass*gravity*9.81). #'Fuel efficiency' shows how much thrust can be extracted from the same mass of fuel with the same engine. For liquid hydrogen + liquid oxygen, it is 1. Flight Simulation Rocket simulations use high-tech computers with complex algorithms. It is impossible to simulate them with very high precision. Still, we can get satisfactory results using Microsoft Excel, with a margin of error around 10%. To make a simulation close to real, we have to figure out what is happening in every second when the engines are firing. At first, the ship will wight much, as it will be filled with fuel. The TWR (thrust to weight ratio) '' will be very small first, but it will rise later. At launch, it is good to have a TWR between 1.2 and 1.5. A too small TWR will make the ship use too much fuel to counter gravity. A too high TWR means the ship will accelerate fast. Air friction will make it lose power and will destroy the fairings protecting the payload. If you try to launch from a planet with a rarefied or no atmosphere, you can have a much higher TWR. If you launch from Venus, which has a dense atmosphere, you will need a small TWR, around 1.1 (and keep it low, to avoid excess friction). A ship is made of several stages. Sometimes, the first stage, fully loaded with fuel, is too heavy. To counter this, the ship might have additional boosters. The first stage has a greater delta-v, but a part of it is lost as the ship fights against gravity. The second stage, which burns outside of the atmosphere, will not have this problem, It will use its force only to increase the speed. Sometimes, ships use ion engines to change from an orbit to another one. In this case, a different calculation needs to be done, because ion engines burn for long, sometimes for many years. Chemical Second Stage This is the most simple way to simulate the work of a rocket. Suppose the rocket is already in space and you want to increase its speed. You need a few parameters before you start working: '''Total mass' - the mass of the stage filled with fuel, plus the payload (mass of all upper stages), in t. Dry mass - the mass of the empty stage plus the payload, in t. Fuel mass - mass of all fuel that will be burned, in t. Engine power - power of the engine in kilonewtons. Engine efficiency - how efficient is the engine burning fuel (Isp, see abve). Number of engines. Time to burn all fuel - displayed in seconds. Fuel consumption per second - the amount of fuel used in a second, in t. Fuel efficiency - for liquid hydrogen + liquid oxygen, it is 1. With these numbers, you can then calculate some parameters, for each flying second: # Column A - time (in seconds). First number is 0 (as the first row will show reference numbers). For the other rows, the formula is: = (previous row)+1. # Column B - current mass (in t). First row displays a reference mass: = (total mass)+((fuel consumption per second)*0.5). The other rows use the next formula: = if((previous row value)>(dry mass),(previous row value)-(fuel consumption per second),0). This way, the mass will decrease with fuel consumption each second. Once we get a value equal with dry mass, we know that there is no fuel and we will get the value zero. # Column C - delta-v (in m/s). This shows the change in velocity in each second. Formula is: = if(mass)>0,(((number of engines)*(engine power))/(current mass))*1.137*(fuel efficiency). Once these calculations are done, you can get the total Delta-v. The formula is: = SUM(c3:c3000). Basically, you need to add all cells on column C displaying delta-v increase in each second. First Chemical Stage The first stage also fights against gravity. Because of this, we need to calculate how much delta-v will be lost to counter gravity. In advance from chemical second stage, we need the following parameter: Gravity (Earth = 1) is the planet's gravity. We also need a few additional columns: # Column D - TWR. The formula is: = ((number of engines)*(engine power))/((current mass)*(gravity)*9.81). ''Remember, for a body with atmosphere, TWR must, at first, not be above 1.5. Also, if TWR is below 1, the ship will go down. # Column E - '''total delta-v'. The formula is: = if((current mass)>0,(delta-v)/(TWR),0). By doing so, you will get the real change of velocity for each second when the engines are firing. After doing this, you will be able to calculate some global parameters: * Crude delta-v = SUM(c3:c3000) or the sum of all values on column C. * Real delta-v = SUM(e3:e3000) or the sum of real delta-v for each second. You will find out that a first stage might have a crude delta-v around 7300 m/s and a real delta-v of 5500 m/s. This is true and shows very clear that a significant amount of fuel is spent to counter gravity. First Stage With Boosters Sometimes, the first stage is too heavy and can lift itself only after a significant amount of fuel is used. In this case, to increase TWR, we have to use boosters. A booster is an attached small rocket that will fire only for a short amount of time, then it will separate from the core and fall back to surface. Some ships might have over ten boosters, be them of the same type or different. How do we calculate the behavior of a first stage with many boosters of various sizes, like those used in Kerbal Space Program? It is complicated, but still can be computed with a Microsoft Excel working sheet. We need to define a few more values: Full mass: the mass of the core stage or of the booster, fully loaded. Dry mass: the mass of the core stage or of the booster, with no fuel. Fuel mass: the mass of fuel available in the core or in the booster. Number of boosters: the number of boosters of the same type. Fuel consumption per second: the amount of fuel used in each second by the core stage or the booster. # Column A - time (in seconds). The first row shows 0, the other ones increase (1,2,3 and so on). # Columns B to G - current mass (t). This will be done for the core stage (column B) and for each type of booster (columns C to G). For the first row, the formula is: = (full mass)*((fuel consumption per second)/2). For the next columns, the formula is: = if((previous row)>(dry mass),(previous row)-(fuel consumption per second),0). This ensures that empty boosters will not count for extra mass, as they will be detached. # Column H - current total mass (t). This adds mass of all parts. The formula is: = ((core stage B current mass))+((booster C current mass)*(number of boosters C))+((booster D current mass)*(number of boosters D))+((booster E current mass)*(number of boosters E))+((booster F current mass)*(number of boosters F))+((booster G current mass)*(number of boosters G))+(payload). # Columns I to N - power (kN). This displays the force each group of boosters contribute to the ship thrust. The formula is: = if((current booster/stage mass)>0,(number of engines)*(engine power)*(number of boosters)*1.137*(fuel efficiency),0). By doing so, you will get the amount of power produced by each group of boosters until they are detached and value will become 0. # Column O - delta-v(m/s). This shows you how much increase in velocity will occur during each second of engine firing without gravity. The formula is as follows: = if((current total mass)>(payload),((power of core stage)+(power of C boosters)+(power of D boosters)+(power of E boosters)+...+(power of G boosters))/(current total mass),0). # Column P - TWR. The formula is: = (combined force of all boosters and core)/((current total mass)*(gravity)*9.81). Keep an eye on this, because you don't want at any moment the TWR to be below 1. The ship will go down. # Column Q - real delta-v (m/s). This shows how much effective delta-v will you get. The formula is: = if((delta-v)>0,(delta-v)-((delta-v)/(TWR)),0). After all this, it is time to finally get the results. Crude delta-v is the sum of all cells on column O. Real delta-v is the sum of all cells on column Q. Ion Engine Stage Ion engines fire sometimes for years. They have a high efficiency (Isp is usually higher then 2000, while chemical engines usually have an Isp between 300 and 450). Because of this, we need to compute their values differently. Time to burn all fuel - will be displayed in days. Fuel consumption per day - instead of fuel consumption per second, displayed in t. Basically, all calculations will be computed like for a second chemical stage, with a few modifications: # Column A - time (in days). First number is 0 (as the first row will show reference numbers). For the other rows, the formula is: = (previous row)+1. # Column B - current mass (in t). First row displays a reference mass: = (total mass)+((fuel consumption per day)*0.5). The other rows use the next formula: = if((previous row value)>(dry mass),(previous row value)-(fuel consumption per day),0). This way, the mass will decrease with fuel consumption each second. Once we get a value equal with dry mass, we know that there is no fuel and we will get the value zero. # Column C - delta-v (in m/s). This shows the change in velocity in each day. Formula is: = if(mass)>0,(((number of engines)*(engine power))/(current mass))*1.137*(fuel efficiency)*86400. Once these calculations are done, you can get the total Delta-v. The formula is: = SUM(c3:c3000). Basically, you need to add all cells on column C displaying delta-v increase in each day. Incomplete Burn Many times, a ship performs an incomplete engine burn. For example, a ship with a ion engine might want to explore two asteroids (like Dawn did). Another example is a ship that plans to land on the Moon and take off with the same fuel tank. In this case, you need to add a new column that will add all delta-v results in previous seconds. This is done in the following way: For a landing/launch mission (which requires calculations like for the first stage): Column G will look like this: * First burn: Suppose values start from cell G3 and you are at cell G135: =SUM(G$3:G135). Scroll down until you get to the cell you want. * Second burn. From the cell where you landed, add (let's suppose it is G135), modify as follow. Let's say you are at cell G227. =SUM(G135:G227). For an orbit mission (like gaining orbit around an asteroid and leaving orbit). No matter if you use a chemical or a ion engine, the column will be D. * First burn: Suppose values start from cell G3 and you are at cell G135: =SUM(G$3:G135). Scroll down until you get to the cell you want. * Second burn. From the cell where you finished maneuver, add (let's suppose it is G135), modify as follow. Let's say you are at cell G227. =SUM(G135:G227). Final Calculations When planning a space mission, it is good to start from the end. So, suppose you want to send a ship to land on Jupiter's moon Europa. The ship will have: *Stage 1 - to leave Earth's atmosphere, with two boosters; *Stage 2 - to get low Earth orbit, chemical engine; *Stage 3 - to travel to Jupiter using a ion engine; *Stage 4 - Jupiter orbit insertion and navigation to Europa, chemical engine; *Stage 5 - lander, chemical engine. You have to calculate all backwards: *Mass of the lander you will send to Europa; *Mass of stage 5; *Mass of stage 4; *Mass of stage 3; *Mass of stage 2; *Mass of stage 1 and boosters. Each following stage will be previous stage's payload. Important Notes These calculations don't include atmosphere friction. In order to do so, you need to calculate how friction will affect any part of the rocket, mainly the nose cone. Friction depends on atmosphere density and speed. Using these formulas, you can see how a rocket will behave like. However, the hardest part is to come with a rocket design to be tested. The fuel efficiency parameter is added by myself. An engine has a specific Isp defined by its builder and is made for a specific fuel. You, however, can sometimes use an engine with a different type of fuel, with little modifications. However, the engine will behave differently, with a higher or smaller Isp. To see how efficient will be an engine with a different type of fuel (assuming it is compatible), I added the fuel efficiency parameter, which is 1 by default. Category:Math